Search results
Results From The WOW.Com Content Network
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
For the reverse implication, it suffices to show that if has all non-negative principal minors, then for all t>0, all leading principal minors of the Hermitian matrix + are strictly positive, where is the nxn identity matrix. Indeed, from the positive definite case, we would know that the matrices + are strictly positive definite.
If the quadratic form f yields only non-negative values (positive or zero), the symmetric matrix is called positive-semidefinite (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and ...
A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero ...
The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite. When A is a real matrix (hence symmetric positive-definite), the factorization may be written =, where L is a real lower triangular matrix with positive diagonal entries.
The exception is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus the inverse of a positive matrix is not positive or even non-negative, as positive matrices are not monomial, for dimension n > 1.
The matrix exponential of a real symmetric matrix is positive definite. Let be an n×n real symmetric matrix and a column vector. Using the elementary properties of the matrix exponential and of symmetric matrices, we have:
More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal 2 × 2 or 3 × 3 matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.